Asymptotic Counting in Conformal Dynamical Systems

In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These results have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known results and proves new results. Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius operators and Tauberian theorems as used in classical problems of prime number theory.

Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels

We study subgame $$\phi $$ ϕ -maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, $$\phi $$ ϕ denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame $$\phi $$ ϕ -maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by $$\phi $$ ϕ . First, we provide necessary and sufficient conditions for a strategy to be a subgame $$\phi $$ ϕ -maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame $$\phi $$ ϕ -maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function $$\phi ^*$$ ϕ ∗ with the following property: if a player has a subgame $$\phi ^*$$ ϕ ∗ -maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame $$\phi $$ ϕ -maxmin strategy for every positive tolerance function $$\phi $$ ϕ is equivalent to the existence of a subgame maxmin strategy.

Expansive dynamics on locally compact groups

Let $\mathcal {G}$ be a second countable, Hausdorff topological group. If $\mathcal {G}$ is locally compact, totally disconnected and T is an expansive automorphism then it is shown that the dynamical system $(\mathcal {G}, T)$ is topologically conjugate to the product of a symbolic full-shift on a finite number of symbols, a totally wandering, countable-state Markov shift and a permutation of a countable coset space of $\mathcal {G}$ that fixes the defining subgroup. In particular if the automorphism is transitive then $\mathcal {G}$ is compact and $(\mathcal {G}, T)$ is topologically conjugate to a full-shift on a finite number of symbols.

Markov chains with exponential return times are finitary

Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of ℤ is a finitary factor of an i.i.d. process.

Regularity of Geometric quadratic stochastic operator generated by 2-partition of infinite points

In this research, we construct a class of quadratic stochastic operator called Geometric quadratic stochastic operator generated by arbitrary 2-partition of infinite points on a countable state space , where . We also study the limiting behavior of such operator by proving the existence of the limit of the sequence through the convergence of the trajectory to a unique fixed point. It is established that such operator is a regular transformation.

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated.